A026374 → θ+1 → x ↦ x/(x+1)

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Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  8,  10;
  3: 1, 11,  31,   25;
  4: 1, 16,  75,  130,    75;
  5: 1, 19, 117,  315,   385,   175;
  6: 1, 24, 195,  740,  1425,  1350,   500;
  7: 1, 27, 258, 1220,  3180,  4650,  3575,  1125;
  8: 1, 32, 370, 2180,  7375, 14900, 17750, 11500,  3125;
  9: 1, 35, 454, 3090, 12535, 31925, 51550, 51250, 28625, 6875;

The row polynomials satisfy (1+4*x+6*x^2+4*x^3+x^4)*P(n,x) = -(1+14*x+81*x^2+254*x^3+476*x^4+550*x^5+385*x^6+150*x^7+25*x^8)*P(n-4,x) + (2+18*x+62*x^2+108*x^3+102*x^4+50*x^5+10*x^6)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 3*x, P(2,x) = 1 + 8*x + 10*x^2, P(3,x) = 1 + 11*x + 31*x^2 + 25*x^3.

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[2, x_] := 1 + 8*x + 10*x^2;
P[3, x_] := 1 + 11*x + 31*x^2 + 25*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + 14*x + 81*x^2 + 254*x^3 + 476*x^4 + 550*x^5 + 385*x^6 + 150*x^7 + 25*x^8)*P[n - 4, x] + (2 + 18*x + 62*x^2 + 108*x^3 + 102*x^4 + 50*x^5 + 10*x^6)*P[n - 2, x])/(1 + 4*x + 6*x^2 + 4*x^3 + x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; v[3] = 1 + 8*x + 10*x^2; v[4] = 1 + 11*x + 31*x^2 + 25*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(1 + 14*x + 81*x^2 + 254*x^3 + 476*x^4 + 550*x^5 + 385*x^6 + 150*x^7 + 25*x^8)*v[i - 4] + (2 + 18*x + 62*x^2 + 108*x^3 + 102*x^4 + 50*x^5 + 10*x^6)*v[i - 2])/(1 + 4*x + 6*x^2 + 4*x^3 + x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from theta_plus_one of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) f…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.